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Full text of "Treatise On Analysis Vol-Ii"

9   MEASURABLE FUNCTIONS       141

(13.9.15)    If fis integrable and if A, B are two measurable sets such that
A n B is negligible, then

f     fdp= {'fdfi + f/d/i.
JAuB           JA          JB

For <pAu B and <pA + <pB are then equivalent functions (12.7.3).

Remark

(13.9.16)    If I is a compact interval [a, b"] in R, then for any regulated
function / with compact support we have f f dX = f f(x) dx, where A is

Jl                Ja

Lebesgue measure (13.8.8). If now/is an arbitrary regulated function defined
on R, then / is measurable because it is the limit of the integrable functions
f<Pin, where ! = [  ,] (13.9.11). For/to be integrable, it is necessary
and sufficient that lim f "\f(x)\ dx < + 00, and in that case we have

n-xxjJ""

(fdA, = lim f+"f(x) dx ((13.8.2) and (13.8.4)); this integral is also denoted

>                  n-+ooj-"

by f *f(x) dx. It can happen that the limit lim f *f(x) dx exists although

J~n                                                        n-^J-n

lim |   |/(jc)| dx = + oo (this is the case, for example, if f(x) = (sin x)/x for

n-*ooj~ 

x ^ 0). The interpretation of most of these "improper integrals" belongs to
the theory of distributions (Section 17.5, Problem 4, and Chapter XXIII).
For the reasons given above, if I c R is any interval with endpoints a, b
(with a :g 6), we write | f(x) dx in place of | /WA, for any function / such
that/cp, is integrable.

The (positive) measure \JL is said to be bounded if the constant function 1 is
integrable, or in other words (13.9.13), if ju*(X) = /**(!) infinite: the number

XX) = n(l) = f d\i is then called the total mass of p (cf. (13,20)).

jx

(13.9.17)    If \JL is bounded, then every bounded measurable function f: X ->> R is
integrable (in particular, every bounded semicontinuous function is integrable)
and

(13.9.17.1)

J/4,

This follows directly from (13.9.13).A/(*) dfi(x);